Astérisque 2011; 291 pp; softcover Number: 335 ISBN10: 2856293050 ISBN13: 9782856293058 List Price: US$90 Member Price: US$72 Order Code: AST/335
 Let \(S\) be a Noetherian separated scheme of finite Krull dimension, and \(\mathcal {SH}(S)\) be the motivic stable homotopy category of MorelVoevodsky. In order to get a motivic analogue of the Postnikov tower, Voevodsky (MR 1977582) constructs the slice filtration by filtering \(\mathcal {SH}(S)\) with respect to the smash powers of the multiplicative group \(\mathbb G_{m}\). The author shows that the slice filtration is compatible with the smash product in Jardine's category \(\mathrm {Spt}_{T}^{\Sigma }\mathcal {M}_{\ast}\) of motivic symmetric \(T\)spectra (MR 1787949) and describes several interesting consequences that follow from this compatibility. Among the consequences that follow from this compatibility is that over a perfect field all the slices \(s_{q}\) are in a canonical way modules in \(\mathrm {Spt}_{T}^{\Sigma }\mathcal {M}_{\ast }\) over the motivic EilenbergMacLane spectrum \(H\mathbb Z\), and if the field has characteristic zero it follows that the slices \(s_{q}\) are big motives in the sense of Voevodsky. This relies on the work of Levine (MR 2365658), RöndigsØstvær (MR 2435654), and Voevodsky (MR 2101286). It also follows that the smash product in \(\mathrm {Spt}_{T}^{\Sigma }\mathcal {M}_{\ast }\) induces pairings in the motivic AtiyahHirzebruch spectral sequence. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in algebra and algebraic geometry. Table of Contents  Preliminaries
 Motivic unstable and stable homotopy theory
 Model structures for the slice filtration
 Bibliography
 Index
